Answer :
To solve the problem of adding the two polynomials [tex]\((7x^6 + 10x^2 - 10)\)[/tex] and [tex]\((3x^6 - 6x^3 + 4)\)[/tex], we'll go through the following steps:
1. Identify like terms: Like terms are those that have the same variable raised to the same power. In this case, we look for terms with [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], and constant terms.
2. Add the coefficients of like terms:
- [tex]\(x^6\)[/tex] terms:
- From the first polynomial, the coefficient is 7.
- From the second polynomial, the coefficient is 3.
- Adding these gives: [tex]\(7 + 3 = 10\)[/tex]. So the [tex]\(x^6\)[/tex] term in the resulting polynomial is [tex]\(10x^6\)[/tex].
- [tex]\(x^3\)[/tex] terms:
- The first polynomial does not have an [tex]\(x^3\)[/tex] term, so its coefficient is 0.
- From the second polynomial, the coefficient is -6.
- Adding these gives: [tex]\(0 - 6 = -6\)[/tex]. So the [tex]\(x^3\)[/tex] term in the resulting polynomial is [tex]\(-6x^3\)[/tex].
- [tex]\(x^2\)[/tex] terms:
- From the first polynomial, the coefficient is 10.
- The second polynomial does not have an [tex]\(x^2\)[/tex] term, so its coefficient is 0.
- Adding these gives: [tex]\(10 + 0 = 10\)[/tex]. So the [tex]\(x^2\)[/tex] term in the resulting polynomial is [tex]\(10x^2\)[/tex].
- Constant terms:
- From the first polynomial, the constant is -10.
- From the second polynomial, the constant is 4.
- Adding these gives: [tex]\(-10 + 4 = -6\)[/tex]. So the constant term in the resulting polynomial is [tex]\(-6\)[/tex].
3. Write the resulting polynomial: Combine all the terms we calculated:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
4. Match the result with the given options:
The resulting polynomial [tex]\(10x^6 - 6x^3 + 10x^2 - 6\)[/tex] matches with one of the answer choices provided.
Thus, the correct answer is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]
1. Identify like terms: Like terms are those that have the same variable raised to the same power. In this case, we look for terms with [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], and constant terms.
2. Add the coefficients of like terms:
- [tex]\(x^6\)[/tex] terms:
- From the first polynomial, the coefficient is 7.
- From the second polynomial, the coefficient is 3.
- Adding these gives: [tex]\(7 + 3 = 10\)[/tex]. So the [tex]\(x^6\)[/tex] term in the resulting polynomial is [tex]\(10x^6\)[/tex].
- [tex]\(x^3\)[/tex] terms:
- The first polynomial does not have an [tex]\(x^3\)[/tex] term, so its coefficient is 0.
- From the second polynomial, the coefficient is -6.
- Adding these gives: [tex]\(0 - 6 = -6\)[/tex]. So the [tex]\(x^3\)[/tex] term in the resulting polynomial is [tex]\(-6x^3\)[/tex].
- [tex]\(x^2\)[/tex] terms:
- From the first polynomial, the coefficient is 10.
- The second polynomial does not have an [tex]\(x^2\)[/tex] term, so its coefficient is 0.
- Adding these gives: [tex]\(10 + 0 = 10\)[/tex]. So the [tex]\(x^2\)[/tex] term in the resulting polynomial is [tex]\(10x^2\)[/tex].
- Constant terms:
- From the first polynomial, the constant is -10.
- From the second polynomial, the constant is 4.
- Adding these gives: [tex]\(-10 + 4 = -6\)[/tex]. So the constant term in the resulting polynomial is [tex]\(-6\)[/tex].
3. Write the resulting polynomial: Combine all the terms we calculated:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
4. Match the result with the given options:
The resulting polynomial [tex]\(10x^6 - 6x^3 + 10x^2 - 6\)[/tex] matches with one of the answer choices provided.
Thus, the correct answer is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]
